238 IX ANALYTIC FUNCTIONS

Now

f(zl9..., Zjfc.i, zft, 0fc+1, ..., ap) — f(zl9..., Zfc-i, ak, ak+1,..., ap)

= Dkf(zl9 ...9zk-l9ak + t(zk - a*), ak+l9...9 ap)(zk - ak) dt.

Write gk(u) =/(zt, . . ., zfc_!, w, flfc+1, . . . , 0P); ft is analytic in an open set
of C containing the ball \u — ak\ ^ r, and |[ft(w)|| < rap in that ball. Applying
(9.9.3) to gk and to the circuit t -*• afc + r*tt defined in [0, 2n]9 we obtain

for \u — flj ^ r/2. Therefore, for any z e Q, and anyfe <& we have

which shows €> is equicontinuous at the point a. The last statement of
(9.13.1) follows from the fact that any bounded set in a finite dimensional
space is relatively compact ((3.17.6) and (3.20.17)), and from Ascoli's
theorem (7.5.7).

(9.13.2) Let A be an open connected set in Cp, <& a set of analytic mappings
of A into a complex Banach space E. Suppose for each compact subset L of A,
the set <I>L of restrictions to L of the functions /e €> is relatively compact in
#E(L). If M. is a set of uniqueness (Section 9.4) in A, and if a sequence (/„) of
functions of ® converges simply in M, then (fn) converges uniformly (to an
analytic function) in any compact subset of A.

From (3.16.4) it follows that we need only prove that, for every compact
set L c A, the sequence of the restrictions of the/n to L has only one cluster
value in ^E(L). Suppose the contrary, and let (#„), (hn) be two subsequences
of (fn)9 each of which converges uniformly in L, the limits being distinct.
As A is locally compact (3.18.4) and separable, there exists an increasing
sequence (UJ of open subsets of A, such that Urt (closure in Cp) be compact
and contained in Un+1, and A = (J Un (3.18.3). Define by induction on k

n

a sequence (#&„)„=1,2,... ? sucn that (gkn) is a subsequence of (ft_i,n), with
9on — 9n -> and that (gkn) converges uniformly in Ufc, which is possible by the
assumption on <J>. Then the "diagonal" subsequence (gnn) converges
uniformly in every Uw, hence, by (9.12.1) its limit g is analytic in A. In a
similar way it is possible to extract from (/*„) a subsequence (hnn) which|| ^rap for all z e P and all/e O. Let Q be the closed ball